Topological spaces - some heavily used invariants - Lec 05 - Frederic Schuller

This is from a series of lectures - "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller

[amirkhan]

I can't believe that I work 12 hours everyday and then spend my evenings watching this virtuoso delivering his masterpiece!

[whdaffer]

I love the fact that you can see him actually working out the implications of what he's trying to teach while he's presenting it. Which so convincingly shows that he's thinking about the proof rather than merely reciting it!

[ndmath]

2:28 at the end definition of T1: I think it should be: ... there is a neighbourhood U ... : q is not a member of U

[tim]

At 30:40 should the definition of refinement be:
For every u-tilde in C-tilde there exists a u in C such that ...

Liking these lectures!

[whdaffer]

A beautiful moment at -31:21 where he asks "hang on, what am I trying to show"

The guy's a genius!

[klausmattis]

The refinement definition should be the other way round: for every U-Twiddle in C-twiddle there exists an U in C s.t. U-twiddle is a subset of U.

Otherwise "subcover => refinement" wouldn't hold

[dev_sda]

Thanks you professor for the great lecture such a treasure to share this series of lectures with us for free ❤

[rdbury]

There were three typos/errors that were written down that I caught, plus one or two places where FS obviously misspoke so not worth noting. First, as mentioned below, the definition of T1 the letter p is used instead of q. (For T0 spaces this is the same as saying singletons are closed.) Second, also as mentioned below, Heine-Borel only applies to subsets of R^n. It can be generalized to metric spaces in general, but you have to add completeness and the subset has to be totally bounded. Third, the definition of the loop space L sub p does not use the letter p anywhere. It should read gamma(0)=gamma(1)=p instead of gamma(0)=gamma(1). Also note that the space is usually assumed to be path connected before defining the fundamental group. That way its isomorphism class does not depend on the choice of p. I\m not entirely convinced that subcover => refinement, but it's probably not important. I haven't caught any errors in the previous videos in the series, so it's surprising that there would be several all at once; I guess FS forgot to eat his Wheaties that morning.

[Gabriel-mfwh]

The Heine Borel theorem does not work for metric spaces in general. The general version is: a subset of a metric space is compact iff it is complete and totally bounded.

Also, there was a typo in the definition of the refinement, it should be
"for every Ũ in the refinement, there is a U in the cover where Ũ is contained in U"
and not
"for every U in the cover, there is a Ũ in the refinement where Ũ is contained in U"
otherwise, subcovers are not necessarily refinements, by that definition, as noted by, for instance



where you can take the top interval out of the cover and it would still be a cover; however, the top, which is in the cover, does not contain any set of the subcover

[zfengjoe]

At 24:34, Heine-Borel should be stated as a property of the topological space instead of a theorem. In a complete metric space, the compactness is equivalent to the closed + totally boundedness (not just boundedness). For R^d and manifold, the boundedness and totally-boundedness are the same, hence we say they have the Heine-Borel property.

[jamesgruthbeard]

Does anyone know if I can find problem sheets to these lectures?

[LaureanoLuna]

At 1:14:00, M\A itself would do the job, i.e. any of A, B can be used to show the existence of some nonempty clopen set other than M.

[taraspokalchuk]

27:39
but there is an infinite subcover (the cover itself.) If the subcover couldn't be equall to the cover than we could construct such cover on [0, 1] that would make it not compact.
for example {[0, 1/2) (1/2, 1] and [1/4, 3/4]}. here the only finite subcover is the cover.

[millerfour]

4:45, 7:27, 16:18, 20:30, 33:32, 39:33, 53:56 (normalization by partition of unity), 1:01:48, 1:07:42, 1:23:56, 1:33:00, 1:36:43, 1:40:08, 1:49:52, 1:54:45

[kerljenge]

38:20 R (in the usual topology) is not homeomorphic to Z x [0, 1). This is easy to check, as Z x [0, 1) is not connected whereas R is.

[KipIngram]

I don't understand why the (x, sin(1/x)) U (0, 0) isn't path connected. (0, 0) is the only point excluded from the (x, sin(1/x)) part, but we add it back in. I understand those are both open sets, but it still looks like we just left out a single point and then re-added it.

[Esloquees]

27:55 (I had to watch it twice) [My mind]:-"Conclusión: Perfect"

[kaiwenwu]

0:09:38 compactness and paracompactness
1:03:06 connectedness

[active]

50:04 for all points in the manifold :).

[neelmodi]

Quick question about the definition of open refinement: The given definition seems to imply that you could have additional covers in the open refinement than in the initial cover. Is this true?

It would seem that a better definition should be: for all "u twidle" in "C twidle" there exists a u in C such that u contains "u twidle"

I might be wrong, but something seems a bit fishy here... Could someone explain?